# The Stacks Project

## Tag 0062

Lemma 5.19.2. Let $X$ be a topological space.

1. Any closed subset of $X$ is stable under specialization.
2. Any open subset of $X$ is stable under generalization.
3. A subset $T \subset X$ is stable under specialization if and only if the complement $T^c$ is stable under generalization.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file topology.tex and is located in lines 3230–3240 (see updates for more information).

\begin{lemma}
\label{lemma-open-closed-specialization}
Let $X$ be a topological space.
\begin{enumerate}
\item Any closed subset of $X$ is stable under specialization.
\item Any open subset of $X$ is stable under generalization.
\item A subset $T \subset X$ is stable under specialization
if and only if
the complement $T^c$ is stable under generalization.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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